Number Theory Questions (9)

If $1(50)^{49} + 2(51)^1(50)^{48} + 3(51)^2(50)^{47} + \cdots + 50(51)^{49} = k(50)^{49}$, then the number of factors of $k$ of the form $4m+1$, ($m \in W$) is/are

Let $X=(23\sqrt3+39)^{2025}$, $Y=X-[X]$ (GIF). Remainder when $XY$ is divided by 31 is

Remainder when $(2022)^{2023}$ is divided by 11 is

If $1(50)^{49} + 2(51)^1(50)^{48} + 3(51)^2(50)^{47} + \cdots + 50(51)^{49} = k(50)^{49}$, then the number of factors of $k$ of the form $4m+1$, ($m \in W$) is/are

Remainder when $(2022)^{2023}$ is divided by 11 is

Let $X=(23\sqrt3+39)^{2025}$, $Y=X-[X]$ (GIF). Remainder when $XY$ is divided by 31 is

Let $X=(23\sqrt3+39)^{2025}$, $Y=X-[X]$ (GIF). Remainder when $XY$ is divided by 31 is

Remainder when $(2022)^{2023}$ is divided by 11 is

If $1(50)^{49} + 2(51)^1(50)^{48} + 3(51)^2(50)^{47} + \cdots + 50(51)^{49} = k(50)^{49}$, then the number of factors of $k$ of the form $4m+1$, ($m \in W$) is/are